Beiträge zur Algebra und Geometrie
Contributions to Algebra and Geometry
Vol. 47, No. 1, pp. 53-62 (2006)
On the perimeter of the intersection of congruent disks}
Károly Bezdek, Robert Connelly and Balázs CsikósDepartment of Mathematics and Statistics, University of Calgary, 2500 University Drive N.W., Calgary, Alberta, Canada T2N 1N4, e-mail: email@example.com; Department of Mathematics, Cornell University, Malott Hall, Ithaca, NY 14853 USA, e-mail: firstname.lastname@example.org; Department of Geometry, Eötvös University, Budapest, P.O.B. 120, H-1518 Hungary, e-mail: email@example.com
Abstract: Almost 20 years ago, R. Alexander conjectured that, under an arbitrary contraction of the center points of finitely many congruent disks in the plane, the perimeter of the intersection of the disks cannot decrease. Even today it does not seem to lie within reach. What makes this problem even more important is the common belief that it would give a sharpening of the well-known Kneser-Poulsen conjecture for the special case of the intersection of congruent disks. Since the Kneser-Poulsen conjecture has just been proved in the plane, we feel that it is a good idea to call attention to this somewhat overlooked conjecture of Alexander. In this note, we prove Alexander's conjecture in some special cases.
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Electronic version published on: 9 May 2006. This page was last modified: 4 Nov 2009.